Research On Paradox In Bayesian Model Selection

Bayesian model selection has important applications in various fields,such as analyse classical hypothesis testing,phylogenetic trees in biology.Therefore,it has received great attention from researchers in many fields such as mathematics,statistics,and biology at home and abroad.But it has been found that in some special cases,the Bayesian model selection will have a paradox phenomenon,which leads to the Bayesian model selection method not applicable to some special models.At present there are four main paradoxes:No free parameters models paradox,Tangent models paradox,Overlapping models paradox and Crossing models paradox.This paper mainly studies the paradox in the overlapping models.Given the alternative model Hk,and the alternative model has overlapping parts.When the true value belongs to the overlapping interval,the K-L distance between alternative model and real model is 0,that is models are equal right.We should choose any model with equal probability 1/k.But by exploring the asymptotic behavior of the posterior model probability,we find that as the increasing of data sets,the posterior model probability of the alternative model converges to a special distribution rather than the expected degenerate distribution 1/k.It mainly proves the existence of paradox in the overlapping model by using the normal distribution and binomial distribution.At the same time,this paper put forward a method which is changing the super-parameter in prior make it relevant to the data size.It appear to be effective to solve the paradox.It is found that when the value of the hyperparameter is greater than 1/2 less than 1,the posterior model probability converges to a single-point distribution 1/k with the data sets increases.Thus the paradox of overlapping model selection is solved.